Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C37 · Schur multiplier (H2) = C24 |
| direct product | C37×#2383 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C37 | C24 |
| commutator | C22 | C23×C4×C37 |
| Frattini, cosocle | C23 | C24×C37 |
| prime radical | C24×C37 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C37 ◁ G | |
456 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 264 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 2368 |
A minimal faithful permutation representation — 4 generators on 57 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)(38,39,40,41)(43,46)(44,48)(50,51,54,52)(53,56,57,55);(43,46)(47,49)(51,52)(53,57);(42,43)(44,47)(45,46)(48,49)(50,52,54,51)(53,56,57,55);(42,44,45,48)(43,47,46,49)(50,53)(51,55)(52,56)(54,57)1406 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C37×#2383, C37×#34350, C4×C37×#560, … |
| maximal normal subgroup of | C2×C37×#2383, C37×#34350, C4×C37×#560, … |
| maximal quotient of | C37×#13830, C37×#13831, C37×#13833, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 1115);